TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HEAT-COVARIANCE-REVIEW.txt Original file SHA-256: 169194868f1ccbe1ca45e4b6166a3ccb65eb286331fde105b6992442e7455989 Inclusive original lines: 1-140 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | SPECTRAL HEAT MIXTURE AND GEOMETRIC TIME: TARGETED ORIGINAL REVIEW 000002 | 29 October 2026. Independent analytic review; no new source/model choice, 000003 | large numerics, original edit or Atlas mutation. 000004 | 000005 | Verdict 000006 | The original proof names a Euclidean COVARIANCE and reflection tau->-tau, 000007 | but never specifies its actual heat-mixture variable or covariance operator. 000008 | A positive Laplace mixture of spectral heat operators does NOT by itself 000009 | establish this geometric-time RP. A single heat covariance has an explicit 000010 | negative reflected2x2 minor. If 'reflection-stable' is intended to mean that 000011 | each component is already OS positive, positive superposition is valid, but 000012 | that stronger property is a premise needing its actual kernel proof. 000013 | 000014 | 1. What the original says, and does not say 000015 | Original archive45 QFT source: 000016 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000017 | 002-section-closure-theorems-from-the-admissibility-complex.tex:267-310, 000018 | registered paper-source:35c4c41ef6cded2368b36527, read directly. 000019 | 000020 | The lemma assumes commutation with admissibility projectors, a bosonic kernel 000021 | with the Bernstein property and reflection stability, and an RP fermionic CAR 000022 | kernel. Its proof:299-306 calls the bosonic object a Euclidean covariance and 000023 | claims a positive superposition of 'reflection-stable heat kernels', followed 000024 | by positivity under reflection across tau=0. No equation C=f(K), integral 000025 | parameter, or factorization into positive-time vectors occurs in these lines. 000026 | 000027 | The cited Bernstein remark in boundary source001:345-350 only calls the 000028 | spectral test profile an analytic realization of an already fixed channel. 000029 | It does not identify the spectral heat parameter with geometric time or 000030 | state that the actual covariance is f(-partial_tau^2+omega^2). That specific 000031 | covariance must therefore not be reported as the computed original TFPT one. 000032 | The exact general inference from spectral heat positivity can nevertheless 000033 | be tested without inventing or replacing the original physical source. 000034 | 000035 | 2. Smallest decisive spectral-heat covariance test 000036 | Fix a single spatial frequency with omega>=0 and geometric time tau on R: 000037 | K=-partial_tau^2+omega^2, 000038 | C_s=exp(-s K), s>0. 000039 | This is a positive covariance operator with a positive point Laplace measure 000040 | at spectral heat parameter s. Its translation kernel is real, positive and 000041 | invariant under tau reflection: 000042 | c_s(u)=exp(-s omega^2)(4pi s)^(-1/2)exp(-u^2/(4s)). 000043 | The reflected covariance on positive times a,b is c_s(a+b), not c_s(a-b). 000044 | For any distinct a,b>0 its2x2 Gram determinant has ratio 000045 | c_s(2a)c_s(2b)/c_s(a+b)^2 000046 | =exp(-(a-b)^2/(2s))<1. 000047 | Thus the determinant is strictly negative despite covariance positivity and 000048 | reflection invariance. For example s=1,a=1,b=2, set 000049 | d=exp(-omega^2)/(2sqrt(pi)), 000050 | f=delta_1-exp(7/4)delta_2. 000051 | Then its reflected quadratic form is exactly 000052 | q(f)=d[exp(-1)-exp(-1/2)]<0. 000053 | The deltas can be replaced by smooth real mollifiers supported in small 000054 | neighborhoods of1 and2 inside tau>0; smoothness of c_s preserves the strict 000055 | negative sign. Hence this is a genuine supported-test-function obstruction, 000056 | not an artifact of allowing delta insertions. 000057 | 000058 | This one atom already disproves sufficiency of an arbitrary positive Laplace 000059 | measure for time RP of f(K). It does NOT assert that every mixture fails: 000060 | integrating heat kernels with suitable weights can yield an RP covariance. 000061 | It is not a complete P1/P2 or admissibility countermodel. 000062 | 000063 | 3. The whole continuous smooth scalar covariance class 000064 | There is an elementary stronger class test, not a new general mathematical 000065 | theorem. Let c:R->R be continuous, even, ordinarily positive definite, with 000066 | c(0)>0. Assume its positive-time Hankel matrices [c(t_i+t_j)] are PSD and 000067 | c is C1 at0. Then c is constant. 000068 | 000069 | Proof: continuity extends the Hankel positivity to a time0 entry and gives 000070 | c(0)c(2t)>=c(t)^2, t>=0. 000071 | Ordinary PD gives |c(t)|<=c(0). Put d(t)=1-c(t)/c(0)>=0. The displayed 000072 | Hankel inequality becomes 000073 | d(2t)<=2d(t)-d(t)^2<=2d(t). 000074 | Iterating yields d(t)<=2^n d(t/2^n). Evenness and C1 at0 imply c'(0)=0, 000075 | hence d(u)=o(u). For each fixed t the right side therefore tends to0, 000076 | so d(t)=0 globally. Evenness covers negative t. Conversely a nonnegative 000077 | constant covariance plainly meets both positivity conditions. 000078 | 000079 | Thus no nonconstant continuous scalar translation-invariant time covariance 000080 | that is differentiable through0 can have both ordinary PD and time RP. In 000081 | particular this applies to any genuinely time-smoothed Gaussian covariance 000082 | with these hypotheses; no scan over individual positive heat weights is 000083 | needed. It does NOT apply to distribution/contact kernels, infinite c(0), 000084 | nontranslation-invariant kernels, nontrivial reflection matrices or graded 000085 | fermionic kernels. Smoothing only spatial coordinates does not imply C1 000086 | through geometric time0. The usual free covariance exp(-E|t|)/(2E) has 000087 | precisely a time cusp at0 and is not excluded. Positive heat mixtures which 000088 | produce such a cusp remain compatible with the sufficient criterion below. 000089 | 000090 | 4. Strong valid statements, with the two parameters kept separate 000091 | An actual positive ENERGY semigroup has a different reflected kernel. For 000092 | H>=0 and vectors v_i, times tau_i>0, 000093 | M_ij= 000094 | =. 000095 | This is a positive Gram matrix. Positive integrable superpositions of such 000096 | already RP kernels remain RP. The heat covariance counterexample does not 000097 | touch this statement, the existing HS heat-word construction, the positive 000098 | coupled excitation semigroup, or their given full Clifford source. 000099 | 000100 | A useful sufficient covariance criterion in the scalar translation-invariant 000101 | class is a positive Stieltjes/resolvent representation 000102 | F(z)=integral_(mu>=0) d rho(mu)/(z+mu), rho>=0, 000103 | with the required convergence and positive frequencies. For 000104 | K=-partial_tau^2+omega^2 it gives 000105 | c(u)=integral d rho(mu) exp(-sqrt(omega^2+mu)|u|) 000106 | /[2sqrt(omega^2+mu)]. 000107 | The reflected c(tau_i+tau_j) is a positive integral of rank-one Gram matrices. 000108 | This is stronger than arbitrary positive Laplace measure in spectral heat 000109 | parameter s. For example the full resolvent integral of exp(-sK) is RP even 000110 | though each individual heat covariance above fails. No choice of such a 000111 | representation for the original source is made here. 000112 | 000113 | 5. Relation to the old finding and exact original obligation 000114 | The Atlas finding positive-spectral-reweighting-does-not-establish-original- 000115 | source-rp and source_sewing_20261029/PROOF.txt:1-184 were reread. Its actual 000116 | counterexample concerns a Boltzmann/Schur reweighting of a reflected measure 000117 | by exp[-Tr f(D^2)]. It does not contain the heat COVARIANCE minor above. 000118 | These are distinct mathematical inferences; neither is a new all-P1 no-go. 000119 | 000120 | For the original lemma, either 'reflection-stable' denotes only covariance 000121 | under reflection, in which case the generic heat-mixture inference is false, 000122 | or it includes reflected positivity itself, in which case that part is an 000123 | explicit hypothesis rather than a derivation from Bernstein weights. The 000124 | first decisive original input is the actual bosonic covariance and its 000125 | reflection, followed by a positive-time kernel factorization or direct RP 000126 | test. The fixed positive Z_ref denominator preserves an already RP form; it 000127 | cannot change the sign of the negative minor. Full joint boson/fermion 000128 | positivity also retains its separate source assumptions. 000129 | 000130 | Primary context, checked directly: Arthur Jaffe, Stochastic PDE, Reflection 000131 | Positivity, and Quantum Fields, https://arxiv.org/pdf/1411.2964 , sectionIII. 000132 | That paper distinguishes auxiliary stochastic time from physical reflection 000133 | time and proves RP failures for its finite-time stochastic covariance. It is 000134 | context for the type distinction, not a claim that its covariance is the 000135 | single heat atom used in this independent elementary calculation. 000136 | 000137 | Given Seed RP, the existing whole Clifford RP intertwiner and all valid 000138 | energy-semigroup Gram proofs remain intact. No original covariance value, 000139 | complete source-map exclusion, physical-time identification or T1-T8 closure 000140 | is asserted here.